Sự tồn tại và hành vi tập trung của các nghiệm trạng thái cơ bản cho một lớp phương trình Schrödinger cận tuyến tổng quát trong ℝN

Acta Mathematica Scientia - Tập 40 Số 5 - Trang 1495-1524 - 2020
Chen, Jianhua1, Huang, Xianjiu1, Cheng, Bitao2, Tang, Xianhua3
1Department of Mathematics, Nanchang University, Nanchang, China
2School of Mathematics and Statistics, Qujing Normal University, Qujing, China
3School of Mathematics and Statistics, Central South University, Changsha, China

Tóm tắt

Trong bài báo này, chúng tôi nghiên cứu phương trình Schrödinger cận tuyến tổng quát $$ - \text{div}({\varepsilon ^2}{g^2}(u)\nabla u) + {\varepsilon ^2}g(u)g'(u){\left| {\nabla u} \right|^2} + V(x)u = K(x){\left| u \right|^{p - 2}}u,\;\;\;\;x \in {\mathbb{R}^N},$$ trong đó N ≥ 3, ε > 0, 4 < 22*, g ϵ C1 (ℝ, ℝ+), V ϵ C (ℝN) ∩ L∞ (ℝN) có điểm cực tiểu toàn cục dương, và K ϵ C(ℝN) ∩ L∞ (ℝN) có điểm cực đại toàn cục dương. Bằng cách sử dụng một sự thay đổi biến, chúng tôi xác định được sự tồn tại và hành vi tập trung của các nghiệm trạng thái cơ bản cho vấn đề này và thiết lập một hiện tượng suy giảm theo cấp số nhân.

Từ khóa

#phương trình Schrödinger #nghiệm trạng thái cơ bản #hành vi tập trung #suy giảm theo cấp số nhân

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